nLab
spherical T-duality

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

String theory

Contents

Idea

Spherical T-duality (Bouwknegt-Evslin-Mathai 14a) is the name given to a variation of topological T-duality where the role of the circle S 1S^1, or the circle group U(1)U(1), is replaced by the 3-sphere S 3S^3, or the special unitary group SU(2)SU(2). Where topological T-duality relates pairs consisting of total spaces of U(1)U(1)-principal bundles equipped with a cocycle in degree-3 ordinary cohomology, spherical T-duality relates pairs consisting of SU(2)SU(2)-principal bundles (or just S 3S^3-fiber bundles (Bouwknegt-Evslin-Mathai 14b)) equipped with cocycles in degree-7 cohomology. As for topological T-duality, under suitable conditions spherical T-duality lifts to an isomorphism of twisted K-theory classes of these bundles with twisting by the 7-class.

While topological T-duality is meant to be a formalization of the T-duality seen in 2d CFT sigma-models/string theory, the physical meaning of spherical T-duality, if any, remains unclear at this point, for the moment it takes its motivation from the fact that it is mathematically possible.

On the other hand, a special role is played in the theory by those SU(2)SU(2)-principal bundles which arise as pullbacks of the quaternionic Hopf fibration along a map from base space to the 4-sphere, and this is, at least rationally, just the structure of the M2/M5-brane charges (Fiorenza-Sati-Schreiber 15, Schreiber 15). So maybe there is a relation…

References

The idea is due to

which in the course considers higher Snaith spectra and higher order algebraic K-theory.

A special case of this general story is discussed in some detail in

Examples in M-theory are discussed in

Last revised on September 7, 2018 at 11:05:56. See the history of this page for a list of all contributions to it.